Local existence and blowing up phenomena for a class of non-autonomous partial functional differential equations with infinite delay

Abstract

In this work, we study a class of abstract non-autonomous partial functional differential equations with infinite delay. Our main results concern the local existence of the mild solution which can blow up at the finite time. The unbounded operators associated to the non-autonomous system are assumed to be stable family which generates C 0-semigroups while the nonlinear part is supposed to be continuous. Under Lipschitz condition on the nonlinear term of the equation, we prove the existence and uniqueness of the mild solution. For illustration, we provide an example for some reaction-diffusion non-autonomous partial functional differential equations involving infinite delay.